Computed FFT cyclic convolution and zero-padded linear convolution

Fix an instance of the operationwise floating-point model in Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, with representable real set $\mathbb{F}$. Input two finite complex sequences $a,b$ whose real and imaginary coordinates all belong to $\mathbb{F}$, a mode cyclic or linear, and phase/FFT controls: an exact rational twiddle tolerance $0<\tau \le 1\mathrm{/}4$, integer dyadic exponent limits $e_{\min }\le e_{\max }$, an initial phase precision $p_0\ge 2$, a finite precision cap $p_{\max }\ge p_0$, a finite elementary-series cap $J\ge 1$, and a maximum radix-two recursion depth $d_{\max }\ge 0$. At precision $p$, phase endpoints use the dyadic format with precision $p$ and exponent limits $e_{\min },e_{\max }$. In cyclic mode the inputs must have the same length $M=2^q$ with integer $q\ge 0$; the requested exact operation is cyclic convolution on $\mathbb{Z}\mathrm{/}M\mathbb{Z}$. In linear mode, with lengths $n,m\ge 1$, set $L=n+m-1$, choose the least power of two $M\ge L$, and append zeros to both inputs to length $M$; the requested output is the first $L$ entries of their cyclic convolution. Validate the mode, positive dimensions, input representability, exact rational tolerance, integer control parameters and their displayed inequalities, and the cyclic power-of-two condition before any transform call. If any such input constraint fails, return inconclusive with diagnostic invalid-input and the offending parameter or coordinate, without a vector output. An arithmetic range failure returns range-failure with its operation and index, without a vector output.

For either mode, call Computed unitary radix-two FFT with certified twiddle phases on each length-$M$ input with direction forward and the same $(\tau ,e_{\min },e_{\max },p_0,p_{\max },J,d_{\max })$ controls. Multiply corresponding transform entries as ${\mathrm{fl}}_{\mathbb{C}}({\widehat{A}}_k{\widehat{B}}_k)$ using the six-real-operation complex product of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add. Call the same FFT routine on this product vector with direction inverse and the same controls, and multiply each output component by the computed real scalar $\mathrm{fl}(\sqrt{M})$ using componentwise rounded real scaling. Return the length-$M$ cyclic result, or its first $L$ entries in linear mode, only if every phase, recursion-budget, and arithmetic operation succeeds; otherwise return the first inconclusive or range-failure status and index. The exact convolution identity and numerical-error claim are separate results.